By Louis Boutet de Monvel, C. De Concini, C. Procesi

Lectures Given on the second consultation of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Venezia, Italy, June 12-20, 1992

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Gr" M]z, associ~ au prerfiier terme de cette suite. /T* V) (modulo l'agrandissement Z " C Z~). I1 reste encore s comparer s l'~l~ment cette image K-th~orique et celle du th~or~me. I1 est assez facile de voir que [M]z a m~me image K-th~orique dans Kz,(T*X) que sa "d~formation au c6ne tangent" [gr v g r M ] c z , off CZ C H x v T * X / Y d~signe le c6ne tangent de Z le long de H; il reste alors s montrer que les 61~ments [grgr ~ M] et [gr ~ g r M ] ont la mfime image K-th6orique dans Kz,(T*X). Ce dernier point n'est pas 6vident parce qu'il met en jeu deux filtrations dont la comparaison n'est pas imm6diate.

2 Let us recall that a category C consists in giving: i) A class of elements, called the objects. ii) For each pairs of objects A, B a set home(A, B) of elements called morphisms. iii) For any 3 objects A, B, C a composition of morphisms: homc( A, B) x home(B, C) ---* home(A, C) These data are subject to the following simple axioms: a) Composition is associative whenever defined. b) For every object A there is a morphism 1A E home(A, A) which behaves as a unit element under compositions: 1Af = f and glA = g, when defined.

Series, Princeton University Press 144 (1951). Verdier J. , Categories d~riv~es, ~tat 0, SGA 489 vol. , 1977, pp. 262-311. Q U A N T U M GROUPS C. S Pisa, C. Procesi Univ. di Roma INDEX Chapter 1 Hopf algebras w Hopf Algebras Categories, functors etc. ~2 Complete reducibility Chapter 2 Finite dimensional representations w w w w Finite dimensional representations of algebras and filtrations Twisted derivations and polynomial algebras Representation theory of twisted derivation algebras Representation theory of twisted polynomial rings Chapter 3 Quantum groups w w w w Some properties of finite root systems Quantum groups Degenerations of quantum groups Poisson structures Chapter 4 The Poisson group H w w w w w w w The quantum group A A universal construction associated to the braid group B as functions on a Poisson group Some Hamiltonian fields The geometry of the quantum coadjoint action Verma modules The center Chapter 5 Roots of 1 w w w w Frobeniusmap Baby Vermamodulesand the degree The center ofUe.